Problem 30 - Algebra A Combined Function

Question

Solve each system of equations by the addition method. If a system contains fractions or decimals, you may want to first clear each equation of fractions or decimals. $$ \left\\{\begin{array}{l} \frac{3}{2} x+4 y=1 \\ 9 x+24 y=5 \end{array}\right. $$

Step-by-Step Solution

Answer

The system of equations has no solution.

Step 1: Clear Fractions

To clear the fractions in the first equation, we multiply every term by 2 to eliminate the fraction. The equation \( \frac{3}{2}x + 4y = 1 \) becomes \( 3x + 8y = 2 \).

Step 2: Align Equations

After clearing the fractions, we now have the system: \( 3x + 8y = 2 \) and \( 9x + 24y = 5 \). Notice that the coefficient of \( x \) in both equations can be made equal by multiplying the first equation by 3.

Step 3: Multiply First Equation

Multiply the first equation \( 3x + 8y = 2 \) by 3 to get \( 9x + 24y = 6 \). The system now is: \[ \begin{cases} 9x + 24y = 6 \ 9x + 24y = 5 \end{cases} \]

Step 4: Subtract Equations

Subtract the second equation from the first: \( (9x + 24y) - (9x + 24y) = 6 - 5 \), which simplifies to \( 0 = 1 \). This indicates that there is no solution to the system, as the equations are inconsistent.

Key Concepts

Addition Method

When solving systems of linear equations, the addition method, also known as the elimination method, is a powerful technique. This method allows you to add or subtract equations to eliminate one of the variables, making it simpler to solve the system. Here's how it works:

• Align the equations vertically to easily see the terms with the same variables.
• Adjust the coefficients of one variable so they are equal in both equations. This often involves multiplying one or both equations by a constant.
• Once those coefficients are the same, add or subtract the equations to eliminate that variable.

After the variable is eliminated, you will have a simpler equation to solve for the remaining variable. This method is particularly useful because it keeps the process straightforward and only requires basic arithmetic operations. However, it's crucial to correctly align and manipulate the equations to avoid errors.

Linear Equations

Linear equations form the basis of systems that can be solved using the addition method. A linear equation in two variables takes the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.
A system of linear equations consists of two or more such equations. Here are key properties of linear equations:

• They represent straight lines when graphed on a coordinate plane.
• The solutions to a system are the points where the lines intersect.
• If two lines are parallel (same slope but different intercepts), the system has no solution.

Understanding these properties is crucial when working with linear equations, as they dictate the behavior of the system and guide how to solve it effectively.

Inconsistent System

The term "inconsistent system" refers to a system of equations that has no solution. This typically occurs when the equations represent parallel lines. In the example given in the step by step solution:
The result of \( 0 = 1 \) implies an inconsistency. This arises because:

• Both equations, though derived from different multiplication steps, are essentially parallel.
• Parallel lines never intersect, hence no shared solution exists.

Recognizing an inconsistent system is a vital part of solving equations because it saves time and avoids pursuing a non-existent solution. In the context of the given system, realizing the lines are parallel immediately indicates an inconsistency.